Copyright JCPDS - International Centre for Diffraction Data 2004, Advances in X-ray Analysis, Volume 47. 200 MINIMIZATION OF ERRORS DUE TO MICROABSORPTION OR ABSORPTION CONTRAST Bradley M. Pederson, Krista J. Schaible, and Ryan S. Winburn * Division of Science, 500 University Ave. W., Minot State University, Minot, ND 58707 ABSTRACT In using X-ray diffraction (XRD) and the Rietveld method there are many factors that affect the results and the interpretation of the data that are collected. One of these is microabsorption. Microabsorption has historically been accounted for using the Brindley correction or mainly ignored. This correction has come under much debate in recent years. This study focuses on examining factors that will aid in minimizing microabsorption effects or more correctly, absorption contrast effects. It also assesses microabsorption effects under various conditions, including absorption characteristics of the sample, absorption characteristics of the internal standard and particle size of the internal standard using different radiation energies (copper, cobalt and iron radiation). The data was examined using the Rietveld refinement method and then compared with and without the Brindley correction. The errors in the results appear to follow a second order polynomial, contrary to the linear trend predicted by Brindley s derivation. It also appears that the radiation source has little effect on this trend for mixtures that do not cross an absorption edge for any phases present. The results should aid in the development of a better methodology for analyzing materials in more complex systems. INTRODUCTION The goal of any quantitative procedure is to obtain the most accurate results possible. Within X- ray diffraction (XRD), there are a number of factors that influence the accuracy of the results. One of the more problematic factors, particularly when dealing with complex mixtures such as coal combustion by-products (CCBs), cements/concretes and geologic materials, is microabsorption. Microabsorption stems from differences in the interactions of each material with the X-ray radiation. Each material will absorb the X-ray radiation to a different extent depending on its linear absorption coefficient for the particular wavelength (energy) of radiation being used. In many complex mixtures the differences in linear absorption coefficients are typically large (for instance, CCBs have linear absorption coefficients ranging from 81cm -1 for quartz to 1153cm -1 for magnetite using copper radiation). Microabsorption is not a new concept in XRD. G.W. Brindley described microabsorption and developed a method for correcting for its effect in quantitative XRD in the mid-1940 s [1]. A detailed discussion of the background relating to this research has been given previously [2]. The quantitative analysis protocol used here utilizes a modified version of an equation used by Taylor [3], as given in Equation 2 [4]: ected Wt. Frac. A = (Wt. Frac. A) τ A N Σ j=1 (Wt. Frac. τ j j) (2)
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Copyright JCPDS - International Centre for Diffraction Data 2004, Advances in X-ray Analysis, Volume 47. 201 there τ A and τ j are the Brindley corrections for phases A and j, respectively. Using the weight fractions calculated in a Rietveld refinement [4], a spreadsheet corrects for microabsorption effects post-refinement. This should, in theory, yield a linear plot when the error due to microabsorption is plotted against the linear absorption coefficient (lac) of the sample (or difference between the linear absorption coefficient of the sample and the material), as discussed in Reference 2. The Brindley correction for microabsorption has since been used in quantitative studies using the Rietveld method [3,5-8]. Initial studies in XRD of CCB materials yielded mixed results as to the importance of microabsorption on the accuracy of the Rietveld method in these systems [5-8]. Additionally, preliminary indications from the Quantitative Analysis Round Robin show that the Brindley correction for microabsorption may indeed reduce the accuracy of results obtained using the Rietveld method [9,10]. This study focuses on microabsorption effects in complex mixtures and their effect on the accuracy of the results obtained using the Rietveld method. Mixtures having a wide range of linear absorption coefficients (both the materials used to make the samples and the samples themselves) were examined in order to gain a better qualitative feel for the effects of microabsorption in complex mixtures. MATERIALS AND METHODS Materials used for this study include CeO 2 (ceria, fluorite structure, Cerac, Inc.), Cr 2 O 3 (Eskolaite, corundum structure, Cerac, Inc.), Al 6 Si 2 O 13 (mullite, Kyoritsu Ceramic Materials CO.), SiO 2 (quartz, US Silica, Inc.), Fe 2 O 3 (hematite, Aldrich.), Fe 3 O 4 (magnetite, Aldrich), silica gel (amorphous, EM Reagents), TiO 2 (rutile, DuPont, R900 grade), TiO 2 (rutile, Alpha Aesar), ZnO (zincite, wurtzite structure, Alpha Aesar), Al 2 O 3 (NIST SRM 676, corundum). The particle size and amorphous content of each material was determined, as shown in Table 1 and their corresponding lac s are shown given in Table 2. The particle sizes of the materials were determined using either X-ray disc centrifuge particle size analysis or by point counting using a scanning electron microscope (SEM). The amorphous content was determined by XRD using either SRM 676, which is undergoing certification for amorphous content [11,12], or rutile (DuPont) as an internal standard. The amorphous content of the rutile was determined using SRM 676. Table 1. Particle Size (µm) and Amorphous Content (%) of Materials Used Material Particle Amorphous Material Particle Amorphous Size Size TiO 2 (R900) 0.35 11.6 TiO 2 0.9 6.7 Cr 2 O 3 0.6 0.1 CeO 2 1.1 4.8 ZnO 0.6 0.7 Al 6 Si 2 O 13 2.1 11.8 Fe 2 O 3 1.4 5 Fe 3 O 4 3.5 26 Al 2 O 3 0.5 4.5 SiO 2 1 5.5
Copyright JCPDS - International Centre for Diffraction Data 2004, Advances in X-ray Analysis, Volume 47. 202 The data were collected using a PANalytical X Pert Pro MPD system. The copper data utilized PANalytical s X Cellerator detector with graphite monochromator. The cobalt data utilized PANalytical s X Cellerator detector with anti-scatter device and an iron filter. The data for both radiation types were collected from 20-80 º2θ with a 0.03º step size using 1º fixed divergence slits and 0.04 radian Söller slits on both the incident and diffracted beam. The total time for each scan was 7.5 minutes. The iron data was collected using the Mini-Prop detector and manganese filter. The data were collected from 20-80 º2θ with a 0.03º step size at 2 seconds per step using 1º fixed divergence and anti-scatter slits, a 0.2mm receiving slit and 0.04 radian Söller slits on both the incident and diffracted beam. The total time for each scan was 1.1 hour. The data was analyzed with the program GSAS [13] as outlined previously [5,6]. Table 2. Individual Phase Linear Absorption Coefficient for Each Radiation Type Phase Qz ZnO Ce Cr 2 O 3 Mu Hm Ma TiO 2 Al 2 O 3 Lac Cu 91 288 2082 948 196 1153 1176 549 124 Lac Co 177 490 3283 1619 N/A N/A N/A 957 235 Lac Fe 141 393 3127 1309 281 279 261 776 187 To help minimize the effects of microabsorption, the internal standard is usually chosen such that it has a lac a) near the mean linear absorption coefficient of the sample or b) near the average of the highest and lowest absorbing phases. Three potential internal standards were examined; two rutile materials (TiO 2 ) with differing particle sizes and alumina (Al 2 O 3 - corundum structure). Rutile was chosen for its intermediate absorption coefficient for Cu radiation, while alumina was selected as the second internal standard because it is the Quantitative XRD Standard Reference Material (SRM) supplied by the National Institute of Standards and Technologies (NIST). While rutile seems a valid choice when using Cu radiation, it has a linear absorption coefficient over twice that of the most highly absorbing phase in CCBs when Fe radiation is used. Alumina, on the other hand, has a low linear absorption coefficient when Cu radiation is used, but falls at a more median value for CCBs when Fe radiation is used. RESULTS AND DISCUSSION The initial trend of a 2nd order polynomial seen in the first four mixtures analyzed using Cu radiation [2] was also seen in the full set of eight mixtures. Not only is this trend apparent in the Cu radiation, but initial results show that the Co and Fe radiation also follow this trend (Figure 1). The differences between radiation sources being a shift of the inflection point for each curve with the change in the linear absorption coefficient for that radiation. The various phases all behaved the same relative to the type of radiation that it was investigated under (2 nd order polynomial behavior). There was also little change in the overall trends due to the internal standard used. The similarities suggest that, at this time, the differences in the internal standard and the type of radiation used do not affect this 2nd order polynomial relationship. The two main concerns with microabsorption are (a) how to minimize microabsorption effects and/or (b) how to correct for its effect. The first concern has been addressed above and in reference 2; the second concern will be addressed here. Does the Brindley correction improve the accuracy of the data? Is this accuracy, or lack there of, when using the correction seen in
Copyright JCPDS - International Centre for Diffraction Data 2004, Advances in X-ray Analysis, Volume 47. 203 Relative Percent Error 0-5 0 Co 500 1000 1500 2000-10 Cu Fe -15-20 -25-30 Mixture lac (cm-1) (a) 30 Co 25 Cu 20 Fe 15 10 5 0-5 0-10 500 1000 1500 2000 (b) Mixture lac (cm-1) Figure 1. Comparison of the relative errors obtained using different radiation types for (a) CeO2, (b) quartz, using small particle size rutile internal standard. different radiation types? Before examining the Brindley correction, one must be certain that one is indeed seeing microabsorption in the definition of Brindley [1]. One of the first criterion for using the Brindley correction as outlined in his paper is that the material must be a medium powder (0.01 <µd < 0.1). The materials used in this study have a range of 0.0091 (quartz) to 0.22 (ceria) using copper radiation. The value for ceria falls outside of the range of Brindley s definition of medium power; however, the correction equation should still yield an approximate correction as the value falls near the lower bound of a coarse powder (0.1 < µd < 1). The second criterion for using the Brindley model for microabsorption is that the materials have polycrystalline particles. Given many materials in use today, this may not be an accurate description; in actuality, single crystallite particles (or particles with a limited number of crystallites) may be encountered. Thus the difficulties encountered may not be microabsorption as defined by Brindley, but rather the more general description: absorption contrast errors. As indicated by the trends exhibited by the errors in the individual phases (2 nd order polynomial versus linear), the Brindley correction may not be describing the errors actually encountered within these systems; i.e. they may be a more general absorption contrast error. However, the correction factor as described by Brindley may still offer some improvement in the accuracy of the results obtained when dealing with materials containing phases with extreme differences in absorption coefficients. The data collected was analyzed both with and without using the Brindley correction as described in Equation (2). The corresponding weighted relative percent errors for all of the mixtures studied using the alumina internal standard are shown in Table 3. As can be see from Table 3, the Brindley correction does improve the accuracy of the results, with all mixtures except mixture 6 with iron radiation showing a decrease in the weighted relative error when the correction is applied. One can also notice that the relative amount of correction is smallest when the linear absorption coefficient of the mixture is closer to that of the internal standard (Mixtures 2, 5, and 6) and the correction seems to normalize the errors over the linear absorption range studied here. A set of synthetic Class F fly ash materials was also investigated in this study. The lac s for these materials have some interesting characteristics depending on the tube type. Unlike the other mixtures studied here, the iron containing phases cross an absorption edge, as indicated by the extreme decrease in their absorption coefficients, when the radiation source is changed from Cu to Fe. This affects the relative position for the lac of the internal standard; i.e., the lac for Relative Percent Error
Copyright JCPDS - International Centre for Diffraction Data 2004, Advances in X-ray Analysis, Volume 47. 204 rutile is near the median when copper radiation is used, while the lac for alumina is near the median when iron radiation is used. Table 3. Comparison of the Weighted Relative Percent Error for Each of the Mixtures With and Without Using the Brindley ection (Alumina Internal Standard). Cu Co Fe Mixture lac (cm -1 ) with w/out lac (cm -1 ) with w/out lac (cm -1 ) with w/out 1 700 17.3 13.1 875 21.1 15.7 1100 16.4 6.3 2 350 16 14.4 475 11.0 8.9 600 11.4 9.0 4 1000 20.6 12.5 1300 29.8 18.8 1600 19.0 8.2 5 200 9.6 9.4 300 7.3 7.1 350 3.9 2.2 6 400 7.7 7.4 550 9.7 8.2 650 8.7 9.5 7 650 13.6 9.8 900 15.1 9.4 1050 18.2 14 8 925 16.9 10.5 1400 23.2 14.1 1500 18.7 12.3 For this study, one mixture was made containing quartz (qz), mullite (mu), hematite (hm), and magnetite (ma). Two fractions of this mixture were mixed with rutile (Alpha Aesar) and alumina, respectively. A third fraction was diluted with silica gel, an amorphous material. This was then fractionated to give two mixtures, one mixed with rutile and the other with alumina. Each of the four resulting samples was run using the conditions outlined previously using both copper and iron radiation. The results of the Rietveld refinements using the resulting data are shown in Tables 4. As seen from these results, the alumina internal standard yields results slightly better than the rutile internal standard; this is contrary to what has been seen previously [5-7]. However, the rutile internal standard used in this study is a larger particle size rutile then used previously. Particle size effects may play a role here as indicated by the larger change upon correction for the rutile internal standard, as shown in Table 4. Table 4 shows the effect of using the Brindley correction on this data. The rutile internal standard appears to give results that are less accurate without the correction, but as noted above, improve upon correction. The alumina, on the other hand, appears to show little change in the accuracy of the results when the Brindley correction is utilized. Table 4. Comparison of ected and Uncorrected Under Different Internal Standards and Radiation Sources with Mixtures with (1) and without (2) Silica Gel. Cu Fe Mixture corr w/o corr corr w/o corr 1-rutile 4.7 5.9 3.6 6.7 1-alumina 3.2 2.3 4.8 4.8 2-rutile 5.6 8.6 2.6 16.6 2-alumina 5.0 6.2 10.1 10.2
Copyright JCPDS - International Centre for Diffraction Data 2004, Advances in X-ray Analysis, Volume 47. 205 CONCLUSIONS The trend in the errors for individual phases appears to follow a 2nd order polynomial with an inflection point near the point at which the linear absorption coefficient of the mixture and internal standard are the same. Initial tests suggest Brindley s theory may not be representing what is being seen, as indicated by the trend of individual errors towards a 2nd order polynomial rather than a linear trend. Initial test do not indicate a significant difference in the overall trends in individual errors with differing radiation sources; i.e., 2nd order polynomial behavior. Smaller particle size internal standards tend to give better overall results than one with a larger particle size. The Brindley correction appears to be more effective at decreasing the overall error for a mixture as the particle size of the internal standard increases. For internal standards with small particle size (~0.5 µm or less), the Brindley correction appears to actually increase the weighted relative error in mixtures with linear absorption coefficients within a given range around that of the internal standard. Comparing alumina and rutile, the rutile does better at moderate linear absorption coefficients, but the errors appear to change more dramatically as the linear absorption coefficient increases. The alumina tends to have higher errors in mixtures with lower to moderate linear absorption coefficients. In complex mixtures with a large amorphous content alumina tends to do better without correcting for the linear absorption coefficient than the larger particle size rutile. ACKNOWLEDGEMENTS We would like to thank the National Science Foundation for the grant that allowed for the purchase of a new powder diffractometer (NSF-CHE-0216468). We would also like to thank Minot State University for funds to help complete this research. REFERENCES [1] Brindley GW. 1945 Phil. Mag. 36: 347 [2] Pederson BM, Gonzalez, RM, Winburn RS, 2003. Minimization of Microabsorption Effects in Complex Mixtures. Adv.X-Ray Anal. accepted [3] Taylor JC. 1991. Powd. Diff 6: 2-9 [4] Winburn R.S., Ph.D. Dissertation, North Dakota State University, 1999. [5] Winburn RS, Grier DG, McCarthy GJ, Peterson RB, 2000. Powd. Diff 15:163-172 [6] Winburn RS, Lerach SL, Jarabek BR, Wisdom M, Grier DG, and McCarthy GJ, 2000. Adv. X-Ray Anal. 42: 387. [7] Gonzalez R.M.; Lorbieke T.D.; McIntyre B.W.; Cathcart J.D.; Brownfield M.; Winburn R.S., Adv. X-Ray Anal., 2002, 45, accepted. [8] Madsen IC, Scarlett NVY, Cranswick LMD, and Lwin T. 2001. J.Appl.Cryst. 34: 409 [9] Scarlett NVY, Madsen IC, Cranswick LMD, and Lwin T, Groleau E, Stephenson G, Aylmore M, Agron-Olshina N. 2002. J.Appl.Cryst. 35: 383 [10] Winburn RS, Von Dreele RB, Hug A, Stephens PW, Cline JC., Denver X-Ray Conference, July 29-August 2, 2002,. abstract [11] Winburn RS, Von Dreele RB, Hug A, Stephens PW, Cline JC. Denver X-Ray Conference, August 4-8, 2003. abstract.